REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

The numbers in brackets are assigned according to the American Mathematical

Society classification scheme. The 1980 Mathematics Subject Classification (1985

Revision) can be found in the December index volumes of Mathematical Reviews.

20(65-01, 65Mxx, 65Nxx].—MYRON B. ALLEN III, ISMAEL HERRERA &

GEORGE F. Pinder, Numerical Modeling in Science and Engineering, Wiley,

New York, 1988, x+418 pp., 24 cm. Price $39.95.

This book seeks to give a unified treatment of numerical modeling by combining

material from continuum mechanics, partial differential equations (PDE's) and nu-

merical methods for PDE's. The book is intended to serve as a text for a year-long

graduate course for engineers, scientists and applied mathematicians. The basic

idea of the book, namely, to first derive the equations from physical principles,

then consider fundamental mathematical questions such as existence and qualita-

tive properties of solutions, and finally devise numerical methods (finite difference

and finite element methods), is of course sound. It should help the student to gain

a better understanding of numerical methods if a proper theoretical background

is given, and the theoretical studies are well motivated if the usefulness and im-

portance is shown of numerical modeling in applications. However, such a unified

approach requires a very careful selection of material to be successful.

The present book starts with a chapter where the basic PDE's in continuum

mechanics (fluid mechanics and elasticity) are derived. Then there follow an in-

troductory chapter on numerical methods and three chapters on steady state (el-

liptic), dissipative (parabolic) and nondissipative (hyperbolic) problems involving

both mathematical and numerical aspects. In the final chapter some nonlinear

problems are considered. The style of the book is informal or "nonmathemati-

cal"; for example, no theorems with precise assumptions are stated. To my taste,

the mathematical level of the text is too low to be really suitable for a graduate

course. For instance, variational principles for Poisson's equation and finite ele-

ments are discussed without the notions of the H1-norrci and completeness, which

seems unnecessarily restrictive, since these concepts are now common also in the

engineering literature. Mostly standard material concerning finite difference meth-

ods is presented in the chapters on elliptic, parabolic and hyperbolic problems. The

book also contains material on finite element methods for these problems, but here

the presentation is not adequate. For instance, in Chapter 2, second-order differen-

tial operators are applied to piecewise linear functions which are only continuous,

and it is stated that the error in the energy norm using piecewise linear basis func-

tions is 0(h2). Such things should be very confusing to the student and defeat

the stated ambition of the authors to increase the understanding of the subject.

©1989 American Mathematical Society0025-5718/89 $1.00 + $.25 per page

761

762 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

Chapter 5 contains a section on finite elements for hyperbolic problems, which is a

very active area of research, where rapid progress has been made in the last years,

for example on problems in several dimensions and nonlinear problems. However,

this development is not touched upon in the present book.

To sum up, I find the basic idea of the book to be very natural but I think

it should be possible to give a presentation which would be more precise math-

ematically and more up-to-date numerically, without becoming more difficult to

read.

Claes Johnson

Department of Mathematics

Chalmers University of Technology

and University of Göteborg

S-412 96 Göteborg, Sweden

21[65-02, 76-02].—OLIVIER PiRONNEAU, Méthodes des Éléments Finis pour les

Fluides, Collection Recherches en Mathématiques Appliquées, Vol. 7, Masson,

Paris, 1988, 199 pp., 24 cm. Price FF 175.00.

This short monograph appears in a relatively new collection edited by P. G.

Ciarlet and J.-L. Lions. Although the collection accepts texts in both French and

English, most titles are in French. Such an outlet for high-quality research is

especially welcome for Ph.D. level teaching where there is a definite lack of this

kind of material. In this respect, the book by O. Pironneau is exemplary. It

presents an excellent summary of classical problems of fluid mechanics, including

a discussion of the physical limitations of models. It also presents a fairly large

number of methods for convection-diffusion problems which are probably the most

difficult ones in fluid simulation at high Reynolds number. The results are in no

way complete, but they leave the reader with a real sense of what the issues are.

The treatment of incompressibility is well done, even if the repertoire of elements

is a little short. There is a treatment of less standard boundary conditions, which

can also be very useful for beginners; this kind of material is alway very hard to

find.

Navier-Stokes equations, both for incompressible and compressible flow, are con-

sidered. Again one should not look for complete results but for a broad picture

sketching the main facts.

The book contains a collection of information which can nowhere else be found

in one place. It should definitely be compulsory reading for beginners in flow simu-

lation. Experienced readers will enjoy the practical presentation and will discover

links between facts and methods they may not have thought about before.

Michel Fortin

Département de Mathématiques

Université de Laval

Cité Universitaire

Quebec, Quebec G1K 7P4, Canada

REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 763

22(65-04, 65M60, 65N30]. I. M. SMITH & D. V. GRIFFITHS, Programming

the Finite Element Method, 2nd ed., Wiley, Chichester, 1988, xii+469 pp., 23^

cm. Price $49.95.

The major objective of this text is to present the reader with an understanding of

how finite element concepts are realized in computer software. This is accomplished

rather effectively by means of a software "building block" technique whereby rel-

atively complex programs may be assembled from a library of relatively simple,

special purpose subroutines, each capable of carrying out some important step in

the finite element analysis process. This device allows the reader a rather clear

view of the overall process, since programs can be written quite compactly making

use of building block subroutines.

The author very thoughtfully starts out by assembling subroutines into relatively

narrow, simple programs and progresses to broader, more complex programs by

modifications and additions.

The building block subroutines are categorized as either "black box" routines or

"special purpose" routines. Black box routines are those associated with standard

matrix handling procedures, and, as the name implies, little explanation of the

bases of these routines is provided. Special purpose routines, on the other hand,

accomplish some significant step in finite element analysis. Although a theoreti-

cal basis of each of these routines is presented, these discussions often seemed too

abbreviated to provide adequate background for the finite element novice. It is

recommended that reading of this text either be preceded by reading of an intro-

ductory finite element text or accompanied by a thorough classroom presentation

of the finite element method.

A number of example programs are presented in the various chapters of the text.

These presentations are enhanced by the availability of a summary of the functions

of each special purpose subroutine in an appendix and a listing of each of these

subroutines in another appendix. The reader may have to bounce back and forth

in these various areas of the text as he/she reads, but this becomes relatively easy

as familiarity with the arrangement increases. It will likely be important to many

readers that all subroutines are available for a nominal charge on tape or diskettes

for many computational environments. Fortran 77 is the language of choice for all

subroutines and programs. Reading of programs and subroutines is made easier by

use of mnemonics for subroutine names as well as variable names.

In general, this book is clearly written and well presented with few typographical

errors or awkward constructions. Some readers may dislike use of examples from

both solid and fluid mechanics. Also some examples make use of inefficient analysis

procedures.

The positive aspects of the book far outweigh the negative aspects. It presents

the broad picture of the basis of finite element software very clearly. The detailed

picture of the basis of various finite element concepts is sometimes left incomplete

or fuzzy from the point of view of the novice.

This book will make a fine addition to the library of anyone interested in fi-

nite element computer programming. It is viewed as complementary to the more

764 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

detailed presentations of the finite element concepts found in other texts.

V. J. Meyers

School of Civil Engineering

Purdue University

West Lafayette, Indiana 47907

23[35L65, 35L67, 65N05, 65N30, 76G15, 76H05].—S. P. SPEKREIJSE,

Multigrid Solution of the Steady Euler Equations, CWI Tract 46, Centrum voor

Wiskunde en Informática (Centre for Mathematics and Computer Science), Am-

sterdam, 1988, vi+153 pp., 24 cm. Price Dfl.24.20.

The subject at hand is the numerical solution of the compressible inviscid equa-

tions of fluid dynamics, the Euler equations. Only steady state solutions are desired,

though knowledge of properties of the unsteady Euler equations is required to de-

rive suitable methods. These equations are of great interest in aerodynamics and

turbomachinery applications. One goal of numerical algorithm designers in the field

of computational fluid dynamics is to produce efficient, robust solvers that can be

used as design tools on a routine basis. The designer must be able to vary some

parameter, for example a geometrical quantity, in a specified parameter set, and

study how the solution varies. The millennium is not yet upon us, though it is

perhaps in sight for problems in two space dimensions.

This compact book (153 pages) presents a large amount of material in a remark-

ably clear and concise fashion. From a derivation of the Euler equations and general

material on the Riemann problem for hyperbolic systems in Chapter 1, the author

moves to a long chapter concerned with upwind discretizations of the Euler equa-

tions. The key ingredient is an approximate Riemann solver; the author prefers

Osher's. All the modern apparatus of upwind schemes is clearly presented here:

numerical flux functions, approximate Riemann solvers, total-variation diminish-

ing schemes, monotonicity, and limiters. Noteworthy is the author's new, weaker,

definition of monotonicity for a two-dimensional scheme, which evades the negative

result of Goodman and Leveque that total-variation diminishing schemes in two

space dimensions are at most first-order accurate. The new definition of mono-

tonicity still rules out interior maxima and minima (for scalar equations) but does

not preclude second-order accuracy. The third chapter gives a very brief descrip-

tion of the multigrid algorithm and shows some numerical solutions of a first-order

accurate discretization. Practitioners in the field have found in the past few years

that multigrid algorithms converge well for first-order accurate discretizations of

steady fluid flow equations. The convergence is helped along by a large amount of

numerical viscosity. Unfortunately, this large amount of numerical viscosity leads to

unacceptable inaccuracy in the computed solutions. Multigrid algorithms applied

directly to second-order discretizations (which would give acceptable accuracy) tend

to converge slowly, if they converge at all. This motivates the fourth and last chap-

ter, in which defect correction is used. This allows one to produce a second-order

accurate solution, even though only problems corresponding to first-order accurate

discretizations need to be solved. It seems that this is the first time defect correc-

tion has been used in the solution of any problem in computational fluid dynamics.

Perhaps the use of defect correction here will spark more interest in the technique.

REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 765

There are just a few, easily corrected, typographical errors in the displayed

equations of this book. More numerous, though perhaps no more numerous than is

the norm nowadays, are the misspelled English words. The book does not claim to

be a comprehensive account of the numerical solution of the Euler equations, nor

of upwind methods for the Euler equations (central differences and the competitors

to Osher's scheme are not discussed). Statements like "Nowadays, finite volume

schemes are almost universally used for shock capturing codes" could be misleading

to a novice. There are many working codes which are based on finite difference

methods and which produce very good results for problems with shocks. The

expert reader who has had trouble with convergence to steady state may look at

the convergence plots in Chapter 4 of this book and wonder if the algorithm would

produce solutions accurate to machine zero if iterated indefinitely. One who wanted

to play the devil's advocate could look at some of the plots in Chapter 3 and claim

they showed a convergence rate which is not independent of grid size (the method

converging slower on finer grids).

These caveats noted, this book is the best single reference for a person trying

to construct a multigrid code for the steady Euler equations. In this reviewer's

opinion, a novice in the field of computational fluid dynamics, armed only with this

book, could produce a working code. This is a tribute to the book's completeness

and clarity.

Dennis C. Jesperson

NASA Ames Research Center

MS 202A-1Moffett Field, California 94035

24(65-02, 65N30].—D. LEGUILLON & E. SANCHEZ-PALENCIA, Computation

of Singular Solutions in Elliptic Problems and Elasticity, Wiley, Chichester, 1987,

200 pp., 24 cm. Price $42.95.

The monograph is focused on the explicit computation of the singular solutions

near corners and interfaces in plane elasticity theory for multimaterials.

These problems are governed by the elasticity system with piecewise constant

coefficients. Indeed, the constitutive law is different in each component of the

material under consideration. The jumps in the coefficients are across the interfaces

between the different components. These interfaces are usually assumed to be

straight, or polygonal, lines. One can also view these problems as transmission

problems across these interfaces.

The general theory shows that the solutions to these problems behave as

rau(9) + more regular terms

in polar coordinates, where r is the distance to the singular point under considera-

tion, which is either one corner of the boundary, or a corner of an interface, or even

a point where an interface meets the boundary.

766 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

The corresponding stresses behave as rQ_1 and therefore may become infinite

as r goes to 0 when Rea < 1. Although this is in clear contradiction with the

assumption that the strains remain small, which legitimizes the linear theory, it

turns out in practice that it provides a good hint of where damage may occur. The

smaller a is, the likelier is the damage.

The general theory of singular solutions is not provided in this monograph. How-

ever, assuming that the data, volume loads and surface loads, vanish in the vicinity

of the corner or singular point under consideration, the authors give a very straight-

forward and illuminating analysis that shows the form of the singular solution there.

The analysis of the case of one single material shows that a is the solution of

a rather complicated transcendental equation. The corresponding transcendental

equations for multimaterials seem quite out of reach by classical analysis. This

is why it is sensible to rely on numerical computation for producing approximate

values of the corresponding a.

Two approaches to the actual calculation of a are described. They are based on

the fact that the above u is the solution of a second-order boundary value problem

for a system of two ordinary differential equations that depend quadratically on a.

In the first approach, the boundary value problem is discretized directly by the

finite element method using piecewise linear functions. The approximate problem

reduces to finding the zeros of a determinant D(a) that depends analytically on a.

The method is quite cheap, yet accurate.

In the second approach, one performs a preliminary reduction of the order with

respect to a, by writing the boundary value problem as a first-order system. Then

one faces the more common problem of finding eigenvalues and eigenvectors to a

boundary value problem for a system of ordinary differential equations. Again,

this is approximated by piecewise linear elements. One is left with the problem of

finding the eigenelements for a large matrix that is nonsymmetric in most cases.

This method is more expensive and requires heavier equipment, but it also allows

one to calculate u approximately.

The whole monograph is very clearly written. Many illustrative examples are

provided. It will certainly be very helpful to engineers.

Pierre Grisvard

Laboratoire de Mathématiques

Université de Nice

F-06034 Nice Cedex, France

25(49-01, 49H05].—M. J. SEWELL, Maximum and Minimum Principles—A

Unified Approach, with Applications, Cambridge Univ. Press, Cambridge, 1987,

xvi+468 pp., 23 \ cm. Price $79.50 hardcover, $34.50 paperback.

This is a text on operational variational methods. It concentrates on how to

associate extremum principles with differential equations and applied problems.

The book requires a minimal mathematical background and provides a large number

of examples. The book has many exercises, so it could be used as a text in a

mathematical methods course.

REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 767

The organization is somewhat unusual. The first chapter treats mostly finite-

dimensional saddle-point problems, with one small section on saddle-point problems

for initial value problems. Chapter 2 is entitled "Duality and Legendre transfor-

mations" and includes some singularity theory and a number of interesting ap-

plications. Again, most of the chapter is finite-dimensional, with some infinite-

dimensional examples interspersed.

The heart of the book is in Chapter 3, where the author shows how to asso-

ciate dual variational problems with a saddle functional. This is done without any

treatment of minimax theorems. Most of the examples involve ordinary differen-

tial equations and include variational inequalities and problems with nonstandard

side conditions. Chapter 4 describes how to derive bounds on linear functionals

of solutions of variational problems and develops some formal methods for finding

variational principles for initial value problems.

The last chapter (Chapter 5) is 90 pages long and is entirely devoted to a theory

of variational principles arising in the mechanics of solids and fluids. Much of

this section reflects the author's own research and is a unique contribution to the

literature.

In the first volume of Courant and Hubert, the calculus of variations was de-

scribed as being calculus on function spaces. Sewell's text provides a calculus, but

not an analysis. It tries to present the theory without using the concepts and tools

of real, or functional, analysis. Such an attitude was necessary in Courant's time. It

is not defensible today. Most of the texts on finite-dimensional optimization theory

referenced here use more analysis than this book does.

Equally surprising for a text in applied mathematics, it does not treat approxi-

mate, or numerical, methods for finding solutions. There is no discussion of second

derivative conditions and no treatment of stability theory. It concentrates entirely

on first derivative conditions and on setting up extremum principles. It does not

study the type of critical point occurring at the solution. For the computational

implementation of variational principles, it is crucial to know the type of a critical

point. Algorithms for computing local minima are based on different considerations

than those for finding saddle points.

This book is not directed at mathematicians. It does not define a Hubert space,

nor does it mention Banach spaces, Lebesgue or Sobolev spaces, or (metric) com-

pleteness. The author ignores many analytical details, and there is no discussion

of existence questions. There is no use of the insights of convex analysis, or of

minimax theory, both of which could have simplified much of the exposition. The

Legendre transformation, described at length in Chapter 2, is a way to compute

the convex conjugate of a given functional in convex analysis. It is the properties

of this conjugate functional, not the Legendre transformation itself, that seem to

be essential in much of duality theory. The author ignores the modern theory of

the calculus of variations and the large volume of literature which has been written

in a more sophisticated mathematical language; especially the contributions of the

French, Italian and Soviet schools. Names that are missing from the bibliography

include those of Brezis, Fichera, Mikhlin and Vainberg.

On the other hand, the examples and applications are of considerable interest.

The book might be an appropriate introduction to these fascinating topics for

768 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

someone, with an interest in applications, who is unwilling to learn elementary

functional analysis. Euler would feel completely at home with this text.

Giles Auchmuty

Department of Mathematics

University of Houston

Houston, Texas 77204-3476

26(65-02, 65Kxx, 90Cxx].—R. FLETCHER, Practical Methods of Optimization,

2nd ed., Wiley, Chichester, 1987, xiv+436 pp., 23 \ cm. Price $53.95.

The development of optimization methods played a particularly vigorous role in

the surge of numerical mathematics following the advent of electronic computing.

Constrained optimization took center stage, first starting in the late 1940's with

the development of linear and nonlinear programming by Dantzig, Chames, Frisch,

Zoutendijk, just to name a few. In the late 1950's Davidon's variable metric method,

its subsequent analysis and independent work by Fletcher and Powell, yielded an

entirely new perspective on the endeavor of unconstrained optimization, which at

the time—hard as it is to comprehend now—was largely considered routine. Both

aspects of optimization have been enriched by progress in numerical linear algebra,

and their methodologies are indeed unthinkable without it.

This flourishing field, as described by one of its foremost pioneers, is the topic

of this meticulously crafted book. It provides a concise, coherent, and no-nonsense

description of the major practical techniques, together with evaluations and rec-

ommendations based on the author's extensive experience. His discussions touch

on many alternatives and extensions to key ideas, placing them also in historical

context. Experience and experiment are stressed throughout, and the text is corre-

spondingly interspersed with illuminating numerical examples. Key theorems are

stated and proved rigorously. A rich variety of exercises at the end of each chapter

deepens the understanding and furnishes important additional detail. The entire

material is supported by a carefully selected, yet complete, bibliography. There

are only very occasional lapses into technical jargon, my favorite being "all pivots

> 0 in Gaussian elimination without pivoting" (page 15). All in all, a highly rec-

ommended reading for those acquainted with the fundamentals of numerical linear

algebra and multivariate calculus.

The division into two parts, on "unconstrained" and "constrained" optimization,

respectively, reflects two different flavors of methodology. Each part had previously

been published as separate volumes [1], [2], with the first part reprinted twice.

The second edition, now combining both volumes, deepens the descriptions of key

subjects in the first part and expands the subject matter of the second. Thus, in the

first part, the Dennis-Moré characterization of superlinear convergence in nonlinear

systems has been included and the important Chapter 2 has been substantially

reorganized. In the second part, a section on "network programming", as well as

a brief discussion of Karmarkar's interior point method, have been added. The

reader will also find many new and stimulating questions in the problem sections.

The second edition offers definitely more than the first edition, while retaining the

focus and the crispness of the previous exposition.

REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 769

Chapters 1 through 6 form the first part. The emphasis is on finding "local"

minimizers using local differential information, explicitly in the form of separately

computed gradients, or implicitly by utilizing difference information. The thorny

problem of determining "global" minimizers is only occasionally mentioned. Chap-

ter 1 contains introductory material. The important Chapter 2 surveys the general

structure of the methods and pays well-placed attention to the principles and al-

gorithms for "line search", a feature in many of the subsequently discussed meth-

ods. Chapters 3 and 4 on Newton, quasi-Newton, and conjugate gradient methods

form the core of the first part on unconstrained optimization. The final two chap-

ters round out that material, featuring among others: trust regions, Levenberg-

Marquardt techniques, the Newton-Raphson method and Davidenko continuation

methods for solving systems of nonlinear equations,' the theory of superlinear con-

vergence.

Chapters 7 through 14 form the second part. Here "constrained optimization"

is first and foremost the optimization of functions, generally excluding the large

field of combinatorial optimization. Chapter 7 is again introductory. Chapter 8

features linear programming with emphasis on the simplex method and some of its

variations such as "product form" and ¿[/-factoring of the basis and Dantzig-Wolfe

decomposition. Lagrange multipliers, first- and second-order optimality conditions,

convexity and duality are introduced in Chapter 9 preparatory to the description

of classical nonlinear programming in Chapters 10 through 12: quadratic program-

ming, linearly constrained optimization, zigzagging, penalty and barrier functions,

sequential quadratic programming, feasible directions. Integer programming is only

briefly discussed, stressing branch-and-bound techniques. It shares Chapter 13

with sections on geometric programming and optimal flows in networks. The final

Chapter 14 offers a unique self-contained treatment of nondifferentiable or, rather,

piecewise differentiable optimization.

The book is a classic and invaluable for the practitioner as well as the student

of the field.

Christoph Witzgall

Center for Computing and Applied Mathematics

National Institute of Standards and Technology

Gaithersburg, Maryland 20899

1. R. FLETCHER, Practical Methods of Optimization, Vol. 1: Unconstrained Optimization, Wiley,

Chichester, 1980.

2. R. FLETCHER, Practical Methods of Optimization, Vol. 2: Constrained Optimization, Wiley,

Chichester, 1981.

27(90-01, 90B99].—PANOS Y. PAPALAMBROS & DOUGLASS J. WILDE, Prin-

ciples of Optimal Design—Modeling and Computation, Cambridge Univ. Press,

Cambridge, 1988, xxi+416 pp., 26 cm. Price $49.50.

For anyone interested in modeling, model building, and in particular, optimiza-

tion models and the interaction between optimization and the modeling process,

this book is a must. It combines classical optimization theory with new ideas of

770 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

monotonicity and model boundedness that provide valuable information for deter-

mining the most efficient and correct formulation of a model. It is an easy-to-read

book with clear, modern notation, well-drawn graphs, and an easy-to-follow orga-

nization. It contains just enough theory and proofs to be complete, without being

overburdened, and follows each concept with examples, applications, and realis-

tically designed engineering design problems. Although aimed at the engineering

design student, it is a valuable addition to the libraries of operations researchers,

economists, numerical analysts, and computer scientists. The engineering flavor

might at first be discouraging, but the discomfort quickly vanishes as the down-to-

earth treatment of the topics comes through.

There are eight chapters in the book. Each has a concise introduction to set the

stage for the concepts to be presented, and a summary to tie all the ideas together

and reinforce what was introduced. The chapters are organized so as to create in

the reader an appreciation for models, from their definition to their use.

Chapter 1 is an excellent introduction to mathematical modeling and the differ-

ent types of models. It defines the design optimization problem and most of the

other ideas and issues that are addressed throughout the book, including feasibility

and boundedness, design space topography, data modeling, solution, and computa-

tion. The concept of a "good" model is introduced (one that represents reality in

a simple but meaningful manner), and the limitations of modeling are also touched

on.

The second chapter, on model boundedness, introduces the concepts of verifica-

tion and simplification, which all too often are overlooked in modeling literature

and practice. The authors stress the importance of applying the techniques of this

chapter before performing any detailed computation, and note that in so doing, one

obtains reductions in model size, tighter formulations, more robust models, and

even formulation error detection. In this, the authors could not be more correct.

As computers and computing become cheaper and more powerful, more researchers

are relying on modeling and computational experiments to test new ideas. Problems

which could not be solved before are now being addressed with success. The ideas

introduced in this chapter are critical to this process and are the same ones used so

successfully, for example, to solve large-scale integer programming problems. (See,

e.g., [1], [2].)

Chapter 3, on interior optima, provides the mathematical foundation for local

iterative methods. Its concise treatment is especially useful for those who want

not only to understand the theory but also to use the methods to solve problems.

Optimality conditions are presented here, along with convexity, gradient methods,

Newton-type methods, and stabilization using modified Cholesky factorization. Un-

fortunately, there is no discussion about interior point methods for solving linear

programming problems.

Boundary optima are the subject of Chapter 4. The discussion proceeds from

feasible directions to sensitivity analysis. Also included are discussions of reduced

gradients, Lagrange multipliers, the generalized reduced gradient method, gradient

projection, and Karush-Kuhn-Tucker conditions. After the general case is treated,

linear programming is developed at the end of the chapter, an approach I found

appealing.

REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 771

In Chapter 5, on model reduction, the ideas from Chapters 2, 3, and 4 are

combined into a well-developed approach to the idea of producing as tight a for-

mulation as is possible. Two new concepts are introduced to help in this search

for a minimal formulation, a rigorous Maximal Activity Principle, and a heuris-

tic Coincidence Rule. I found this theoretical approach to model reduction very

appealing.

Global bound construction, the topic of Chapter 6, is the logical next step in

the progression toward developing useful models. Constructing tight bounds, as

the authors note, not only saves effort, but avoids accepting suboptimal solutions.

Many techniques for constructing bounds are introduced here, from simple lower

bounds to geometric inequalities, to unconstrained geometric programming. Com-

bining these techniques into a process for model reduction using branch and bound

is also discussed.

Chapter 7, on local computation, contains descriptions of various numerical al-

gorithms for solving nonlinear programming problems. It is introduced with a brief

section on choosing a method from among the large number of generally accepted

techniques that are available. It contains sections on convergence, termination cri-

teria, single variable minimization, Quasi-Newton methods, differencing, scaling,

active set strategies, and Penalty and Barrier methods.

The real action is in Chapter 8, Principles and Practice. It begins with a review

of the modeling techniques and approaches presented in the previous chapters. The

goal is "to point out again the intimacy between modeling and computation that

was explored first in Chapter 1." An optimization checklist is also provided for the

novice modeler to use as a prompt while gaining more experience.

Richard H. F. JacksonCenter for Manufacturing Engineering

National Institute of Standards and Technology

Gaithersburg, Maryland 20899

1. K. L. HOFFMAN & M. W. PADBERG, Techniques for Improving the Linear Programming

Representation of Zero-One Programming Problems, George Mason Univ. Tech. Rep., 1988.

2. H. CROWDER, E. L. JOHNSON & M. PADBERG, "Solving large-scale zero-one linear pro-

gramming problems," Oper. Res., v. 31, 1983, pp. 803-834.

28(65-01, 65Fxx].—Dario Bini, Milvio Capovani & Ornella Menchi,

Metodi Numerici per l'Algebra Lineare, Zanichelli, Bologna, 1988, x+514 pp., 24

cm. Price L. 48 000 paperback.

An excellent treatment of numerical methods in linear algebra, this text is des-

tined to become the standard work on the subject in the Italian language. It

contains seven chapters, of which the first three are introductory, providing the

necessary tools of matrix algebra, eigenvalue theory and estimation, and norms.

Chapter 4 discusses the major direct methods, Chapter 5 the more important iter-

ative methods, for solving linear algebraic systems. Methods for computing eigen-

values and eigenvectors of symmetric and nonsymmetric matrices are the subject of

Chapter 6. The final chapter deals with least squares problems and related matters.

772 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

The exposition, throughout, is crisp and to the point. There is a healthy balance

between theoretical analysis, the study of error propagation and complexity, and

practical experimentation. Each chapter is provided with a superb collection of

exercises—many supplied with solution sketches—and with historical notes and

bibliographies. For future editions, however, the authors may wish to pay more

attention to the correct spelling of names.

W. G.

29[68T99, 65D07, 65D10].—ANDREW BLAKE & ANDREW ZlSSERMAN, Vi-

sual Reconstruction, The MIT Press Series in Artificial Intelligence, The MIT

Press, Cambridge, Mass., 1987, ix+225 pp., 23 \ cm. Price $25.00.

This book appears in a series on artificial intelligence, and seems to be aimed

primarily at computer scientists. However, mathematicians interested in computer

vision will certainly want to look at it, and it may be of some peripheral interest to

numerical analysts and approximation theorists interested in optimization and/or

splines.

The subject is visual reconstruction, which is defined by the authors to be the

process of reducing visual data to stable descriptions. The visual data may be

thought of as coming from photoreceptors, spatio-temporal filters, or from depth

maps obtained by stereopsis or optic rangefinders. Stability in this setting refers to

the desire that the representation should be invariant to certain distortions such as

sampling grain, optical blurring, optical distortion and sensor noise, rotation and

translation, perspective distortions, and variation in photometric conditions.

The bulk of the book is devoted to the use of certain variational methods (called

here weak strings, weak rods, and weak plates) for detecting edges (discontinuities

in value or slope) of functions and surfaces. The methods are a form of penalized

least squares, where the penalty includes a measure of smoothness of the function

(typically an integral of a derivative) which is reminiscent of spline theory. These

problems are analyzed using variational methods, and solved by discretizing them

and applying an appropriate optimization method. The optimization method dis-

cussed here is referred to as the graduated non-convexity algorithm, and is designed

to work on the nonconvex problems arising here. Convergence properties and the

optimality of the algorithm are discussed.

The book includes about 150 references, mostly in the computer science liter-

ature. The authors seem to assume that the reader is familiar with much of this

literature; results are often referred to without explanation. Readers not familiar

with such things as "pontilliste depth map", "hyperacuity", "cyclopean space",

or "data-fusion machine" may find the going hard. The authors have elected to

"avoid undue mathematical detail", and despite several appendices, I expect that

most mathematicians will not be fully satisfied.

L.L. S.

REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 773

30[01A45]. A. W. F. EDWARDS, Pascal's Arithmetical Triangle, Oxford Univ.

Press, New York, 1987, xii+174 pp., 24 cm. Price $37.50.

As an impressive culmination of meticulous research into original sources, this

definitive study constitutes the first full-length history of the Arithmetical Triangle,

reputedly the most celebrated of all number patterns. Its origins are herein traced

to Pythagorean arithmetic, Hindu combinatorics, and Arabic algebra.

The elements of this triangle evolved historically in three equivalent forms;

namely, figúrate numbers, combinatorial numbers, and binomial coefficients. The

different aspects of these numbers were first combined by Blaise Pascal in his Traité

du triangle arithmétique, written in 1654 and printed posthumously in 1665. He was

the first to consider the properties of this array as pure mathematics, deduced from

the fundamental addition relation independently of any binomial or combinatorial

application.

Dr. Edwards discusses in detail this creative treatise and shows, in particular,

how it influenced Wallis, Newton, and Leibniz in the early development of analysis.

Special attention is given to Wallis's ingenious derivation of his well-known infinite

product by means of interpolation in an analogous triangular array of reciprocals

of figúrate numbers.

The modern theory of probability originated in the extensive correspondence

between Pascal and Fermât concerning the previously unsolved problems called,

respectively, the Problem of Points and the Gambler's Ruin. The solutions obtained

by Pascal and Fermât have been described in detail in two previously published

papers by the author, which are herein reprinted as appendices. Dr. Edwards

establishes that the basic solution of the first problem is properly attributable to

Pascal instead of Fermât, contrary to tradition.

The origins of the binomial and multinomial distributions are also traced to

Pascal's treatise, as revealed in the correspondence of Leibniz, John Bernoulli,

De Moivre, and Montmort.

A concluding chapter is devoted to a discussion of James Bernoulli's famous

work, Ars conjectandi, which is chiefly noted for containing the first limit theorem

in probability. Apparently, James Bernoulli independently derived the binomial

distribution, unaware of Pascal's treatise, to which his attention was first drawn

just before his death in 1705 by Leibniz.

Each of the ten chapters of this carefully written book is supplemented by ex-

planatory notes and bibliographic references. In addition, there is a comprehensive

list of nearly two hundred sources and an index of the names of more than one

hundred persons appearing in this historical account.

This scholarly book should be of special interest to students and teachers of the

history of mathematics and statistics.

J. W. W.

774 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

31[65F10, 65N20].—A. HADJIDIMOS (Editor), Iterative Methods for the Solu-

tion of Linear Systems, North-Holland, Amsterdam, 1988, 291 pp., 27 cm. Price

$78.00/Dfl. 160.00.

This is a reissue in book form of Journal of Computational and Applied Math-

ematics, v. 24, 1988, nos. 1 & 2. It provides a useful cross section of current

work in the area of the title, including topics such as convergence theory, parame-

ter selection, preconditioning, rectangular and infinite systems, implementation on

machines with parallel architectures, and software packages.

W. G.

32(65-06, 68-06].—JAMES McKENNA & ROGER TEMAM (Editors),

ICIAM '87: Proceedings of the First International Conference on Industrial

and Applied Mathematics, SIAM, Philadelphia, 1988, xx+376 pp., 26 cm. Price

$56.50.

The conference in the title of these proceedings, co-organized by four sister soci-

eties of applied mathematics, GAMM, IMA, SIAM and SMAI from Germany, Eng-

land, the United States and France, was held in Paris on June 29—July 3, 1987.

Part I contains the welcoming and opening addresses. Sixteen invited papers make

up Part II. The authors and their titles are: Wolfgang Alt, Modelling of motility in

biological systems; Karl Johan Aström, Stochastic control theory; Michael Atiyah,

Topology and differential equations; Robert Azencott, Image analysis and Markov

fields; Philippe G. Ciarlet, Modeling and numerical analysis of junctions between

elastic structures; D. G. Crighton, Aeronautical acoustics: mathematics applied to

a major industrial problem; Carl de Boor, What is a multivariate spline?; Yves

Genin, On a duality relation in the theory of orthogonal polynomials and its ap-

plication in signal processing; W. Hackbusch, A new approach to robust multi-grid

solvers; John Hopcroft, Model driven simulation systems; Peter D. Lax, Mathemat-

ics and computing; L. Lovász, Geometry of numbers: an algorithmic view; Andrew

Majda, Vortex dynamics: numerical analysis, scientific computing, and mathemat-

ical theory; F. Natterer, Mathematics and tomography; P. Perrier, Numerical flow

simulation in aerospace industry; M. J. D. Powell, A review of algorithms for nonlin-

ear equations and unconstrained optimization. A similar variety of topics is treated

in 69 minisymposia whose abstracts appear in Part III. A list of contributed papers

in Part IV, and an author index, conclude the volume. Full-length versions of 29

contributions by Dutch authors have previously been published in [1].

W. G.

1. A. H.P. VAN DER BURGH & R. M.M. Mattheu (Editors), Proceedings of the First In-

ternational Conference on Industrial and Applied Mathematics (ICIAM 8T): Contributions from the

Netherlands, CWI Tract, Vol. 36, Centre for Mathematics and Computer Science, Amsterdam,

1987. [Review 21, Math. Comp., v. 50, 1988, p. 649.]

REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 775

33(65-06]. ROLAND GLOWINSKI, G ENE H. GOLUB, GÉRARD A. MEURANT

&¿ JACQUES PeriAUX, First International Symposium on Domain Decompo-

sition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988,

x+431 pp., 26 cm. Price $48.50.

This is the proceedings of a symposium on the subject of the title of this volume.

It consists of 22 papers by invited speakers. The papers presented were not subject

to the refereeing process. The topics of the articles include theoretical foundations,

applications to physical problems, related techniques such as block relaxation and

computer implementation of decomposition algorithms.

J.H.B.

34(33-06, 33A65, 41-06, 42C05].—M. ALFARO, J. S. DEHESA, F. J.

MARCELLAN, J. L. RUBIO DE FRANCIA & J. VINUESA (Editors), Orthog-

onal Polynomials and Their Applications, Lecture Notes in Math., vol. 1329,

Springer-Verlag, Berlin, 1988, xv+334 pp., 24 cm. Price $28.60.

These are the proceedings of the Second International Symposium on Orthogo-

nal Polynomials and their Applications held in Segovia, Spain, September 22-27,

1986. (For the first symposium, see [1].) The volume contains nine invited lectures,

covering analytic and group-theoretic aspects of orthogonal polynomials and appli-

cations to approximation theory. In addition, there are 13 contributed papers and

a collection of open problems.

W.G.

1. C. Brezinski, A. Draux, A. P. Magnus, P. Maroni & A. Ronveaux (eds.), Polynômes

Orthogonaux et Applications, Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985. (Review

23, Math. Comp., v. 49, 1987, pp. 305-306; Corrigendum, ibid., v. 50, 1988, p. 359.)

35(65-06, 68-06].—GARRY RODRIGUE (Editor), Parallel Processing for Scien-

tific Computing, SIAM, Philadelphia, 1989, 428 pp., 25 \ cm. Price $45.00.

These are the proceedings of the Third SIAM Conference on Parallel Process-

ing for Scientific Computing held in Los Angeles, California, December 1-4, 1987.

The papers (and in some cases abstracts only) of 14 invited lectures and 76 con-

tributed talks are arranged in seven parts: Matrix Computations, Numerical Meth-

ods, Differential Equations, Scientific Applications, Languages, Software Systems,

and Architectures.

W. G.